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MATH 365 MEASURE THEORY AND PROBABILITY

创建日期：2024-04-24 16:08:38

所属分类：数学mathematics

标签： MATH 365

MEASURE THEORY AND PROBABILITY

MATH 365
DEPARTMENT: Mathematical Sciences
MEASURE THEORY AND PROBABILITY
Time allowed: See instructions on Vital
INSTRUCTIONS TO CANDIDATES: This assessment carries 50% towards the
overall module mark. The paper contains 6 questions. Full marks will be given
for complete answers to four questions. Only the best four answers will be taken
into account.
By submitting solutions to this assessment you affirm that you have read and
understood the Academic Integrity Policy detailed in Appendix L of the Code of
Practice on Assessment.
Paper Code MATH 365 Page 1 of 3 CONTINUED
1. Consider the probability space (Ω = [−1,+1],F = B[−1,+1], P = 12m).
(i) Construct the σ-field G generated by the event A = [0, 1]. [7 marks]
(ii) Consider the random variable
X =
{
1, if ω ∈ [0, 1];
0, if ω ∈ [−1, 0)
and construct the σ-field FX generated by X. Are the σ-fields G and FX inde-
pendent? Justify your answer. [10 marks]
(iii) Calculate E(X|G). [8 marks]
2. Let Ω1 = Ω2 = [0, 1] and consider the following function on Ω1 × Ω2:
f(x, y) =

1
x3
, if 0 < y < x < 1;
− 1
2y3
, if 0 < x < y < 1;
0 otherwise.
(i) Calculate
∫
Ω2
(∫
Ω1
f(x, y) dx
)
dy. [10 marks]
(ii) Calculate
∫
Ω1
(∫
Ω2
f(x, y) dy
)
dx. [10 marks]
(iii) Does the statement of the Fubini Theorem hold? Give explanations.
[5 marks]
3. (a) Prove that, for any two sets, A = B if and only if A4B = ∅.
[10 marks]
(b) For the denumerable sequence of sets A1, A2, . . ., we define the sets E and
F as follows
E =
∞⋃
n=1
∞⋂
m=n
Am; F =
∞⋂
n=1
∞⋃
m=n
Am.
(i) For particular case of A1 = {0}, A2 = {0, 1}, A3 = {0}, A4 = {0, 1},
A5 = {0}, A6 = {0, 1}, . . ., calculate E and F . [5 marks]
(ii) Prove that, in general, for any n ∈ N,
En =
∞⋂
m=n
Am ⊂
∞⋂
N=n
∞⋃
m=N
Am.
[10 marks]
Paper Code MATH 365 Page 2 of 3 CONTINUED
4. Let (Ω,F , P ) be a probability space and let ξt be a sequence of mutually
independent random variables on it with
∫
R ξt(ω)P (dω) = 0 and
∫
R ξ
2
t (ω)P (dω) =
σ2t < ∞ for all t = 1, 2, . . .. Let {Ft} be the natural filtration for the process
ξt(ω) and consider the random process
Xt =
(
t∑
i=1
ξi
)2
−
t∑
i=1
σ2i .
Is X an {Ft}-martingale, or sub-(super-)martingale. Justify your answer.
[25 marks]
5. Consider the measurable space (Ω = [0,∞),B[0,∞)). Let Fn be the minimal
σ-field containing all the intervals [0, 1), [1, 2), . . . , [n− 1, n), n = 1, 2, . . ..
(i) What kind of subsets of Ω are elements of Fn ? [15 marks]
(ii) Is {Fn, n ≥ 1} a filtration? Justify your answer. [10 marks]
6. Suppose Ω = (0, 1), F is the σ-field of Lebesgue-measurable subsets of Ω,
and m is the Lebesgue measure. Consider the following sequence of functions on
Ω:
f1(x) = 1;
f2(x) =
{
2, if x ∈ (0, 1
2
);
0, if x ∈ [1
2
, 1);
f3(x) =
{
0, if x ∈ (0, 1
2
];
2, if x ∈ (1
2
, 1),
and so on:
f4(x) equals four only for x ∈ (0, 14); otherwise it is zero,
f5(x) = 4 only for x ∈ (14 , 24), f6(x) = 4 only for x ∈ (24 , 34), f7(x) = 4 only for
x ∈ (3
4
, 1),...
if i = 2n, 2n + 1, . . . , 2n+1 − 1 then
fi(x) =
{
2n, if x ∈ ( i−2n
2n
, i+1−2
n
2n
);
0, otherwise.
Here n = 0, 1, 2, . . .
(i) For an arbitrary x ∈ Ω, compute lim supi→∞ fi(x) and lim infi→∞ fi(x).
[13 marks]
Are the following statements correct?
(ii) limi→∞ fi(x) = 0 almost everywhere on Ω. [4 marks]
(iii) fi converges to zero (as i→∞) in measure. [4 marks]
(iv) fi converges to zero (as i→∞) in Lp-norm, p = 1, 2, . . . . [4 marks]
In all these cases, you should justify your answer.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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